We present equations of motion for the Su-Schrieffer-Heeger (SSH) Hamiltonian derived with the help of ansatz states similar to Davydov's so-called mod D 1 ) state for soliton dynamics in proteins. Such an ansatz state allows for quantum effects in the lattice and goes beyond previous calculations which mostly apply adiabatic models. In the most general case, called mod Phi 0 ), which is treated here in detail, we assume that the coherent-state amplitudes for the lattice depend on the site and the molecular orbital of the electrons. The equations of motion are derived from the Lagrangian of the system, a method which is equivalent to the time-dependent variational principle. In the resulting equations we find that, although the SSH Hamiltonian is a one-particle operator, indirect electron-electron interactions are present in the system which originate from the electron-phonon interactions. Inclusion of direct electron-electron interactions, will give insight into the interplay between electron-electron and electron-phonon interactions which can lead effectively to an attractive force between the electrons in systems other than polyacetylene, where bipolarons are known to be unstable. Further with our time-dependent wavefunction also vibrational details of absorption spectra can be computed. From the equations of motion several approximations can be derived. In a further approximation, mod Phi 2 ), the dependence of the coherent-state amplitudes on the lattice site is neglected. This mod Phi 2 ) ansatz state consists of a simple product of the electronic and the lattice wavefunctions; however, the electrons are not constrained to follow the lattice dynamics instantaneously as in the adiabatic case. Finally the classical adiabatic case is discussed on which soliton-dynamics simulations are usually based. Further we discuss how to include temperature effects in our model. Applications to soliton dynamics are discussed for the example of the mod Phi 2 ) model with emphasis on the dependence of the results on soliton width and temperature. We found that in contrast to results reported in the literature, where a similar ansatz is used, but only one electron is treated explicitly, the solitons remain stable also for small soliton widths. This indicates that the interactions of the electrons not occupying the soliton level with the lattice have a stabilizing effect on the soliton. Further our results indicate that the temperature model using random forces and dissipation terms to introduce temperature effects has to be applied with extreme care in this case due to the strong electron-lattice interactions.
We present equations of motion for the Su-Schrieffer-Heeger (SSH) Hamiltonian derived with the help of ansatz states similar to Davydov's so-called mod D 1 ) state for soliton dynamics in proteins. Such an ansatz state allows for quantum effects in the lattice and goes beyond previous calculations which mostly apply adiabatic models. In the most general case, called mod Phi 0 ), which is treated here in detail, we assume that the coherent-state amplitudes for the lattice depend on the site and the molecular orbital of the electrons. The equations of motion are derived from the Lagrangian of the system, a method which is equivalent to the time-dependent variational principle. In the resulting equations we find that, although the SSH Hamiltonian is a one-particle operator, indirect electron-electron interactions are present in the system which originate from the electron-phonon interactions. Inclusion of direct electron-electron interactions, will give insight into the interplay between electron-electron and electron-phonon interactions which can lead effectively to an attractive force between the electrons in systems other than polyacetylene, where bipolarons are known to be unstable. Further with our time-dependent wavefunction also vibrational details of absorption spectra can be computed. From the equations of motion several approximations can be derived. In a further approximation, mod Phi 2 ), the dependence of the coherent-state amplitudes on the lattice site is neglected. This mod Phi 2 ) ansatz state consists of a simple product of the electronic and the lattice wavefunctions; however, the electrons are not constrained to follow the lattice dynamics instantaneously as in the adiabatic case. Finally the classical adiabatic case is discussed on which soliton-dynamics simulations are usually based. Further we discuss how to include temperature effects in our model. Applications to soliton dynamics are discussed for the example of the mod Phi 2 ) model with emphasis on the dependence of the results on soliton width and temperature. We found that in contrast to results reported in the literature, where a similar ansatz is used, but only one electron is treated explicitly, the solitons remain stable also for small soliton widths. This indicates that the interactions of the electrons not occupying the soliton level with the lattice have a stabilizing effect on the soliton. Further our results indicate that the temperature model using random forces and dissipation terms to introduce temperature effects has to be applied with extreme care in this case due to the strong electron-lattice interactions.
We present a simulation method to compute the time-dependent properties of a system of fermions coupled to classical degrees of freedom. The method is employed to calculate the static and dynamic properties of the Su-Schrieffer-Heeger model for polyacetylene chains. Our results for the static quantities for chains, containing up to 256 electrons on 256 sites, are in excellent agreement with previous work and confirm that solitons are only present for very small doping. Our simulation data for the optical absorption as a function of dopant concentration are in good agreement with experiment.
Characterizing bipolaron binding, and understanding how it depends on electron-phonon interaction, is crucial to unraveling the nature of emergent many-body states in strongly interacting electron-phonon systems. So far, most studies of bipolarons have been limited to the Holstein model, in which the coupling constant is momentum-independent. The paradigmatic example of momentum-dependent electron-phonon interaction comes from the system in which phonon distortions modify electron hopping, the SSH model. Already individual polarons in the SSH model are richer than the Holstein model counterparts, and feature a phase transition into the finite momentum ground state with increasing electron-phonon interaction. In this paper, we use a variational approach to study bipolarons in the one-dimensional SSH model and discuss their ground state, dispersion, and excitation spectra. We explore the full parameter range of the system, including the adiabatic regime of slow phonons, which was inaccessible to previous theoretical studies. In agreement with earlier studies, we find that in the anti-adiabatic strongly interacting regime, bipolarons have low effective mass. By contrast, in the adiabatic case, we find that increasing electron-phonon interactions results in an exponential increase of the bipolaron mass. We establish the existence of multiple branches of bound excited states of SSH bipolaron and discuss the signatures of these bound states in dynamics. We show that in the anti-adiabatic regime, response functions obey a parity selection rule, that imposes symmetry constraints on the excitation spectra and provides a clear signature of SSH bipolarons.
The dynamics of a coupled electron–boson system is investigated by employing a multitude of the Davydov D 1 trial states, also known as the multi-D 1 Ansatz, and a second trial state based on a superposition of the time-dependent generalized coherent state (GCS Ansatz ). The two Ansätze are applied to study population dynamics in the spin-boson model and the Holstein molecular crystal model, and a detailed comparison with numerically exact results obtained by the (multilayer) multiconfiguration time-dependent Hartree method and the hierarchy equations of motion approach is drawn. It is found that the two methodologies proposed here have significantly improved over that with the single D 1 Ansatz, yielding quantitatively accurate results even in the critical cases of large energy biases and large transfer integrals. The two methodologies provide new effective tools for accurate, efficient simulation of many-body quantum dynamics thanks to a relatively small number of parameters which characterize the electron–nuclear wave functions. The wave-function-based approaches are capable of tracking explicitly detailed bosonic dynamics, which is absent by construct in approaches based on the reduced density matrix. The efficiency and flexibility of our methods are also advantages as compared with numerically exact approaches such as QUAPI and HEOM, especially at low temperatures and in the strong coupling regime.
We investigate the two-dimensional optical Su-Schrieffer-Heeger (SSH) model, in which the electron hopping amplitude is modulated by the difference between neighboring phonon coordinates. Using sign-problem-free auxiliary-field quantum Monte Carlo simulations, complemented by mean-field analysis, we determine the long-range ordered phases as a function of the electron-phonon coupling and phonon frequency. By examining both adiabatic and antiadiabatic regimes, we reveal the emergence of staggered and armchair valence bond solid (VBS) phases, as well as the O(4) antiferromagnetic phase. In addition, finite-temperature simulations show that the VBS transition occurs at critical temperatures significantly higher than in models with local electron-phonon coupling, consistent with the presence of lighter polarons in the metallic regime. These findings establish the ground-state and finite-temperature phase diagrams of the optical SSH model, which emphasize its similarities and contrasts with other electron-phonon systems.
Polarons with different types of electron-phonon coupling have fundamentally different properties. When the dominant interaction is between the electron density and lattice displacement, the momentum of the ground state does not change and the polaron gets exponentially heavy at strong coupling. In contrast, one-dimensional Peierls/Su-Schrieffer-Heeger (PSSH) polarons with interaction originating from displacement-modulated hopping feature a shift of the ground-state momentum to finite values and moderate values of effective mass as coupling is increased [D. J. J. Marchand et al., Phys. Rev. Lett. 105, 266605 (2010)]. Based on Diagrammatic Monte Carlo method, we investigate whether unusual properties of PSSH polarons depend on the type of the displacement-modulated hopping and to what degree they survive in higher dimension. We study two different PSSH models: with bosonic degrees of freedom residing on sites (model A) and bonds (model B) of the two-dimensional square lattice. For model A, we find that in both adiabatic and intermediate regimes, the momentum of the ground state experiences a continuous transition from zero to a finite value as a function of coupling strength. The transition is driven by quadratic instability of the dispersion function, implying that effective mass diverges at the critical point, and then decreases in an anisotropic fashion with increasing coupling. Unexpectedly, for model B, the momentum of the ground state always stays at zero and the effective mass increases monotonously with coupling. The increase is far from exponential and tends to level off at strong interaction, resulting in relatively light polarons. Having light polarons in the strong coupling regime is crucial for the bipolaron mechanism of high-temperature superconductivity [J. Sous, M. Chakraborty, R. V. Krems, and M. Berciu, Phys. Rev. Lett. 121, 247001 (2018).]
This Perspective presents an overview of the development of the hierarchy of Davydov's Ansätze and a few of their applications in many-body problems in computational chemical physics. Davydov's solitons originated in the investigation of vibrational energy transport in proteins in the 1970s. Momentum-space projection of these solitary waves turned up to be accurate variational ground-state wave functions for the extended Holstein molecular crystal model, lending unambiguous evidence to the absence of formal quantum phase transitions in Holstein systems. The multiple Davydov Ansätze have been proposed, with increasing Ansatz multiplicity, as incremental improvements of their single-Ansatz parents. For a given Hamiltonian, the time-dependent variational formalism is utilized to extract accurate dynamic and spectroscopic properties using Davydov's Ansätze as its trial states. A quantity proven to disappear for large multiplicities, the Ansatz relative deviation is introduced to quantify how closely the Schrödinger equation is obeyed. Three finite-temperature extensions to the time-dependent variation scheme are elaborated, i.e., the Monte Carlo importance sampling, the method of thermofield dynamics, and the method of displaced number states. To demonstrate the versatility of the methodology, this Perspective provides applications of Davydov's Ansätze to the generalized Holstein Hamiltonian, variants of the spin-boson model, and systems of cavity-assisted singlet fission, where accurate dynamic and spectroscopic properties of the many-body systems are given by the Davydov trial states.